RMS Voltage — Sine, Square, Triangle & Sawtooth

In 2026, a power-quality analyst auditing a data-centre's 480 V three-phase mains needs to convert oscilloscope peak readings into true RMS without assuming a sine. This calculator does exactly that — and shows the wave-integration animation so junior engineers stop confusing "peak÷√2" with a universal formula.

The conceptual root is James Prescott Joule's 1841 paper at the University of Manchester: power dissipated in a resistor is P = I²R, no matter what the waveform looks like. Joule's work made it inevitable that the meaningful average for an AC current was the one that preserved this heating identity — what we now call the root-mean-square. Joule's contemporary James Clerk Maxwell folded the same identity into electromagnetic field theory in 1873, defining mean-square field intensities for radiation.

Heinrich Hertz's 1887–1888 experimental confirmation of Maxwell's electromagnetic-wave prediction made AC ubiquitous — and Charles Proteus Steinmetz at General Electric in 1893 needed a clean formalism to talk about it. His AIEE paper introduced the j-omega complex-number representation and codified Vrms as the engineering norm for sinusoidal AC. By 1900 the IEEE forerunner (AIEE) had standardised RMS notation in its Transactions.

For sixty years, AC meters were thermal — a tiny heater warmed a thermocouple whose DC output was proportional to V²rms by Joule's identity. The Hewlett-Packard 3400A (1965) was a milestone thermal-converter RMS voltmeter. Then in 1983 Fluke shipped the Fluke 8060A — the first handheld "true-RMS" DMM with a digital squarer-integrator-rooter computed on a custom IC. The Fluke 87V (1995, revised through 2024) and 289 (2006) extended true-RMS to crest factors of 3:1 at full scale and 6:1 at one-third scale — exactly what modern SMPS line currents demand. The IEEE 1241-2010 ADC standard codifies the ENOB and SINAD assumptions every true-RMS firmware relies on.

The IEC 60050-101 vocabulary defines RMS formally; IEC 61672 specifies the same math for sound-level meters; IEEE 1459-2010 generalises it to distorted polyphase power systems. The mathematics is identical to the bottom-panel animation on this page — square v(t), integrate over T, divide by T, take √. The widget's four waveform shapes (sine, square, triangle, sawtooth) cover the four canonical Fourier-decomposable test signals; every other repeating waveform is a sum of these and your true-RMS meter handles each harmonic in firmware.

The Westinghouse–Edison "AC wars" of 1886–1893 ultimately ended when Tesla's rotating-field motor and Stanley's 1885 transformer made AC distribution unbeatable — and Steinmetz's RMS algebra made it billable. Today the 120 Vrms US mains and 230 Vrms EU mains (IEC 60038) are the same RMS that Joule's 1841 heating law would predict. Every Tektronix and Keysight oscilloscope's cursor labelled "Vrms" runs the same integration this widget shades blue.

What does the answer really mean? A Vrms of 120 V means a resistive heater hooked to that AC source dissipates exactly the same power as it would on a 120 V DC battery — 120²/R watts. The peak is √2 higher, but the average squared voltage matches. That equivalence is what lets electricians size wire by RMS current per NEC 310.16 ampacity tables and what lets a true-RMS Fluke 87V give a meaningful reading on a square-wave inverter output. Time-averaged squared physics — that is what RMS is.